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Some explicit formulas for the matrix exponential

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2 Author(s)
D. S. Bernstein ; Dept. of Aerosp. Eng., Michigan Univ., Ann Arbor, MI, USA ; W. So

Formulas are derived for the exponential of an arbitrary 2×2 matrix in terms of either its eigenvalues or entries. These results are then applied to the second-order mechanical vibration equation with weak or strong damping. Some formulas for the exponential of n×n matrices are given for matrices that satisfy an arbitrary quadratic polynomial. Besides the above-mentioned 2×2 matrices, these results encompass involutory, rank 1, and idempotent matrices. Consideration is then given to n×n matrices that satisfy a special cubic polynomial. These results are applied to the case of a 3×3 skew symmetric matrix whose exponential represents the constant rotation of a rigid body about a fixed axis

Published in:

IEEE Transactions on Automatic Control  (Volume:38 ,  Issue: 8 )